cp's OEIS Frontend

A207361 Displacement under constant discrete unit surge.

%I A207361 #56 May 25 2022 09:16:06
%S A207361 0,1,11,53,173,448,994,1974,3606,6171,10021,15587,23387,34034,48244,
%T A207361 66844,90780,121125,159087,206017,263417,332948,416438,515890,633490,
%U A207361 771615,932841,1119951,1335943,1584038,1867688
%N A207361 Displacement under constant discrete unit surge.
%C A207361 Assume discrete times 0, 1, 2, 3, ...
%C A207361 Assume constant discrete unit surge (= jerk = rate of change of acceleration) starting surge(0) = 0.
%C A207361 Also assume acceleration(0) = velocity(0) = displacement(0) = 0.
%C A207361 So at t = 0, 1, 2, 3, 4, ... the acceleration = 0, 1, 2, 3, 4, ...
%C A207361 Then the velocity v(t) = v(t-1) + a(t)*t.
%C A207361 So the displacement = s(t) = s(t-1) + v(t)*t.
%C A207361 v(0,1,2,3,4,...) = 0, 1, 5, 14, 30, 55, 91, 140, ... = A000330(n).
%C A207361 The subsequence of primes is finite with three terms 11, 53, and 173.
%H A207361 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A207361 a(0) = 0; for n>0, a(n) = a(n-1) + n*A000330(n) = a(n-1) + n*(0^2 + 1^2 + 2^2 + ... + n^2) = a(n-1) + n^2*(n+1)*(2*n+1)/6 = n*(1+n)*(2+n)*(1 + 11*n + 8*n^2)/120 = (2*n + 25*n^2 + 50*n^3 + 35*n^4 + 8*n^5)/120.
%F A207361 G.f.: x*(2*x^2+5*x+1) / (x-1)^6. - _Colin Barker_, May 06 2013
%F A207361 a(n) = Sum_{i=0..n-1} A108678(i). - _J. M. Bergot_, May 02 2018
%F A207361 a(n) = Sum_{0<=i<=j<=n} i^2*j. - _Robert FERREOL_, May 24 2022
%e A207361 s(4) = s(3) + v(4)*4 =  53 +  30*4 =  53 + 120 =  173;
%e A207361 s(5) = s(4) + v(5)*5 = 173 +  55*5 = 173 + 275 =  448;
%e A207361 s(6) = s(5) + v(6)*6 = 448 +  91*6 = 448 + 546 =  994;
%e A207361 s(7) = s(6) + v(7)*7 = 994 + 140*7 = 994 + 980 = 1974.
%p A207361 a:=n->sum(sum(i^2*j,j=i..n),i=0..n): seq(a(n),n=0..30); # _Robert FERREOL_, May 24 2022
%t A207361 a[0] = 0; a[n_] := a[n] = a[n-1] + n^2*(n+1)*(2*n+1)/6; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Oct 22 2015 *)
%o A207361 (Maxima) A207361(n) := block(
%o A207361         n*(1+n)*(2+n)*(1+11*n+8*n^2)/120
%o A207361 )$ /* _R. J. Mathar_, Mar 08 2012 */
%Y A207361 Cf. A000330, A101856, A108678.
%K A207361 nonn,easy
%O A207361 0,3
%A A207361 _Jonathan Vos Post_, Feb 18 2012